User:XaosBits/Formal

Formal definition

There are two formal definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor. The measure theoretical definitions assumes the existence of a measure-preserving transformation. This appears to exclude dissipative systems, as in a dissipative system a small region of phase space shrinks under time evolution. A simple construction (sometimes called the Krylov-Bogolubov theorem) shows that it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.

Many different invariant measures can be associated to any one evolution rule. In ergodic theory the choice is assumed made, but if the dynamical system is given by a system of differential equations the appropriate measure must be determined. Some systems have a natural measure, such as the Lioville measure in Hamiltonian systems, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For many dissipative chaotic systems the choice of invariant measure is technically more challenging. The measure needs to be supported on the attractor, but attractors have zero Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure.

For hyperbolic dynamical systems, the SRB measures appear to be the natural choice. They are constructed on the geometrical structure of stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.

The difficulty in constructing the natural measure for a dynamical system makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure.

Geometrical definition

A dynamical system is the tuple $\langle {\mathcal {M}},f,{\mathcal {T}}\rangle$, with ${\mathcal {M}}$ a manifold (locally a Banach space or Euclidean space), ${\mathcal {T}}$ the domain for time (non-negative reals, the integers, ...) and f ^t a diffeomorphism of the manifold to itself.

Measure theoretical definition

See main article measure-preserving dynamical system.

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet $(X,\Sigma ,\mu ,\tau )$. Here, X is a set, and Σ is a topology on X, so that $(X,\Sigma )$ is a sigma-algebra. For every element $\sigma \in \Sigma$, μ is its finite measure, so that the triplet $(X,\Sigma ,\mu )$ is a probability space. A map $\tau :X\to X$ is said to be Σ-measurable if and only if, for every $\sigma \in \Sigma$, one has $\tau ^{-1}\sigma \in \Sigma$. A map τ is said to preserve the measure if and only if, for every $\sigma \in \Sigma$, one has $\mu (\tau ^{-1}\sigma )=\mu (\sigma )$. Combining the above, a map τ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The quadruple $(X,\Sigma ,\mu ,\tau )$, for such a τ, is then defined to be a dynamical system.

The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates $\tau ^{n}=\tau \circ \tau \circ \ldots \circ \tau$ for integer n are studied. For continuous dynamical systems, the map τ is understood to be finite time evolution map and the construction is more complicated.